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Автомобиль на перекрестке может поехать прямо, а может свернуть направо или налево. Вероятность попадания в «пробку» при проезде прямо равна $0.5$; направо – $0.3$; налево – $0.2$. Определить вероятность беспрепятственного проезда._ -*Решение*: Пусть $A_1$ - машина едет прямо, $A_2$ - машина поворачивает направо, $A_3$ - машина поворачивает налево. +*Решение*: Пусть направления выбираются равновероятно: -Эти 3 события +$ +P("без пробки") = 1/3 (1 - 0.5 + 1 - 0.3 + 1 - 0.2) = 0.667 +$ -*Ответ*: +*Ответ*: #box()[$0.667$] #line(length: 100%) _3. В торговую фирму поставляются телевизоры тремя фирмами в соотношении 5:2:3. Телевизоры не требуют ремонта в течение гарантийного срока соответственно в 96%, 92% и 94% случаев. Найти вероятность того, что купленный телевизор не потребует ремонта в течение гарантийного срока._ +*Решение*: + +$ +P(F_1) = 5/(10) = 0.5, space.quad P(F_2) = 2/(10) = 0.2, space.quad P(F_3) = 3/(10) = 0.3 +$ + +Надежности: + +$ +P(A | F_1) = 0.96, space.quad P(A | F_2) = 0.92, space.quad P(A | F_3) = 0.94 +$ + +где A - не потребует ремонтажа. + +Тогда + +$ +P(A) = sum_i P(F_i) P(A | F_i) = 0.5 dot 0.96 + 0.2 dot 0.92 + 0.3 dot 0.94 = 0.946 +$ + +*Ответ*: #box()[$0.946$] + #line(length: 100%) _4. Имеются три одинаковых ящика. В первом лежат 2 белых и 2 черных шара; во втором — 3 черных шара; в третьем — 1 черный и 5 белых шара. Некто случайным образом вынимает шар из наугад выбранного ящика. Какова вероятность, что шар будет белый?_ +*Решение*: Три ящика, выбираются равновероятно — значит, $P(B_1) = P(B_2) = P(B_3) = 1/3$. + +Состав: + +#table(columns: 5, inset: 10pt)[*Ящик*][*Белые*][*Чёрные*][*Всего*][$P("белый" | B_i)$][1][2][2][4][$frac(2,4) = frac(1,2)$][2][0][3][3][0][3][5][1][6][$5/6$] + +По формуле полной вероятности: + +$ +P("белый") = sum_(i = 1)^3 P(B_i) P("белый" | B_i) +$ + +$ +P("белый") = 1/3 (1/2 + 0 + 5/6) = 4/9 +$ + +*Ответ*: #box(inset: (x: 2pt, y: 4pt))[$4/9$] + #line(length: 100%) _5. В трех одинаковых урнах находятся шары: в первой с номерами от 10 до 25, во второй от 26 до 32 и в третьей от 33 до 45 включительно. Из случайно выбранной урны берется шар. Какова вероятность того, что его номер будет простым числом?_ +*Решение*: + +Обозначим $A$ — "номер простое число", $U_1, U_2, U_3$ — выбор $1$-й, $2$-й, $3$-й урны. + +Тогда по формуле полной вероятности: +$ +P(A)= 1/3 (P(A | U_1) + P(A | U_2) + P(A | U_3)). +$ + +По формуле условной вероятности: + +Урна 1: числа $10 dots 25$. Простые: $11,13,17,19,23$. + +$ +P(A | U_1) = frac(5, 16) +$ + +Урна 2: числа $26 dots 32$. Простые: $29,31$. + +$ +P(A | U_2) = 2/7 +$ + +Урна 3: числа $33 dots 45$. Простые: $37,41,43$. + +$ +P(A | U_3) = 3/(13) +$ + +$ +P(A) = 1/3 (5/(16) + 2/7 + 3/(13)) approx 0.2763 +$ + +*Ответ*: #box()[$0.2763$] + #line(length: 100%) -_6. Берут две колоды по 36 карт. Из первой колоды во вторую перекладывают 2 карты. Затем из второй колоды берется одна карта. Какова вероятность того, что это дама?_ +_6. Берут две колоды по $36$ карт. Из первой колоды во вторую перекладывают $2$ карты. Затем из второй колоды берется одна карта. Какова вероятность того, что это дама?_ + +*Решение*: Во второй колоде изначально $4$ дамы из $36$ карт. Из первой колоды перекладываем $2$ случайные карты. В первой колоде доля дам $4/(36)$, значит ожидаемое число дам среди переложенных двух равно + +$E[K] = 2 dot 4/(36) = 2/9$ + +Тогда ожидаемое число дам во второй колоде после перекладки: + +$ +4 + E[K] = 4 + 2/9 = (38)/9 +$ + +После перекладки во второй колоде $38$ карт, и мы вытягиваем одну наугад. Безусловная вероятность вытянуть даму равна + +$ +E[frac(4 + K, 38)] = frac(E[4 + K], 38) = frac(4 + 2/9, 38) = frac((38)/9, 38) = 1/9 +$ + +*Ответ*: #box(inset: (x: 2pt, y: 4pt))[$1/9$] #line(length: 100%) _7. В альбоме 7 негашеных и 6 гашеных марок. Из них наудачу извлекаются 2 марки, подвергаются гашению и возвращаются в альбом. После чего вновь извлекают 3 марки. Определить вероятность того, что все 3 марки чистые._ +*Решение*: В альбоме изначально $7$ чистых и $6$ гашеных, всего $13$. Достаем $2$ наугад, гасим их и возвращаем. Пусть $К$ - сколько из этих двух были чистыми. Тогда $K in {0, 1, 2}$. + +$ +P(K = 0) = frac(C_7^0 C_6^2, C_(13)^2) = frac(15, 78) = frac(5, 26), \ +P(K = 1) = frac(C_7^1 C_6^1, C_(13)^2) = frac(42, 78) = frac(7, 13), \ +P(K = 2) = frac(C_7^2 C_6^0, C_(13)^2) = frac(21, 78) = frac(7, 26) +$ + +После этого в альбоме все те же $13$ марок, но чистых осталось $7 - K$. + +Теперь тянем $3$ марки. Условная вероятность, что все $3$ - чистые, при фиксированном $K = k$: + +$ +P("все 3 чистые" | K = k) = frac(C_(7 - k)^3, C_(13)^3) +$ + +Вычислим нужные комбинации: + +$ +C_(13)^3 = 286, space.quad C_7^3 = 35, space.quad C_6^3 = 20, space.quad C_5^3 = 10 +$ + +Значит: + +$ +P("все 3" | K = 0) = frac(35, 286), \ +P("все 3" | K = 1) = frac(20, 286), \ +P("все 3" | K = 2) = frac(10, 286) +$ + +По формуле полной вероятности: + +$ +P = sum_(k = 0)^2 P(K = k)P("все 3" | K = k) = 0.0706 +$ + +*Ответ*: #box()[$0.0706$] + #line(length: 100%) _8. Среди трех игральных костей одна фальшивая. На фальшивой кости шестерка появляется с вероятностью $1/3$. Бросили две игральные кости. Определить вероятность того, что выпали две шестерки._ + +*Решение*: $3$ кости: $2$ честные, $1$ фальшивая. + +У честной кости: $P(6) = 1/6$ + +У фальшивой кости: $P(6) = 1/3$ + +Пусть $C$ - честная кость, $F$ - фальшивая. + +Возможные пары: + +#table(columns: 3, inset: 10pt)[*Пара брошенных костей*][*Вероятность выбора*][*\# шестерок*][C, C][$frac(C_2^2, C_3^2) = 1/3$][обе честные][C, F][$frac(C_2^1 C_1^1, C_3^2) = 2/3$][одна честная, одна фальшивая] + +Если обе честные: + +$ +P(6, 6 | C, C) = (1/6)^2 = 1/(36) +$ + +Если одна честная и одна фальшивая: + +$ +P(6, 6 | C, F) = 1/6 dot 1/3 = 1/(18) +$ + +Формула полной вероятности: + +$ +P(6, 6) = P(C, C) dot P(6, 6 | C, C) + P(C, F) dot P(6, 6 | C, F) = \ += 1/3 dot 1/(36) + 2/3 dot 1/(18) = 0.0463 +$ + +*Ответ*: #box()[$0.0463$]